   Definition 2.10 [1]}
Let  \(n \,\in \, \mathbb {N}\)  and  \(X\)   be a linear space of dimension greater than or equal to  \(n\)  over the field  \(\mathbb {K}\) , where  \(\mathbb {K}\)   is the real or complex numbers field.A n-inner product on  \(X\)   is a map
\(\left(\,x \,,\, y \,,\, x_{\,2} \,,\, \cdots \,,\, x_{\,n}\,\right) \,\longmapsto \, \left<\,x \,,\, y \;|\; x_{\,2} \,,\, \cdots \,,\, x_{\,n} \,\right>,\; x,\, y,\, x_{\,2},\, \cdots ,\, x_{\,n} \,\in \, X\) 
